A mountain climber, in the process of crossing between two cliffs by a rope, pauses to rest. She weighs 575 N. As the drawing shows, she is closer to the left cliff than to the right cliff, with the result that the tensions in the left and right sides of the rope are not the same. Find the tensions in the rope to the left and to the right of the mountain climber.
To find the tensions in the rope to the left and right of the mountain climber, we need to understand the forces acting on her and apply the principles of equilibrium.
Let's consider the forces acting on the mountain climber. The two main forces at play are her weight (acting downwards) and the tension in the rope (acting horizontally).
First, let's calculate the weight of the climber. Given that her weight is 575 N, we can assume this force acts directly downward.
Now, let's analyze the forces acting on the climber when she is at rest. Since she is closer to the left cliff, we'll separate the analysis for the left and right sides of the rope.
On the left side of the rope:
- The tension in the rope pulls to the right.
- The weight of the climber pulls downwards.
On the right side of the rope:
- The tension in the rope pulls to the left.
- The weight of the climber pulls downwards.
Since the climber is at rest, the net force acting on her must be zero in both the horizontal and vertical directions.
By setting up an equilibrium equation in the horizontal direction, we can find the tension on the left side of the rope:
T_left - T_right = 0 (since the net force in the horizontal direction is zero)
By setting up an equilibrium equation in the vertical direction, we can find the tension on the right side of the rope:
T_left + T_right = 575 N (since the weight of the climber equals the sum of the tensions)
Now, we have a system of two equations. Solving these equations simultaneously will give us the tensions on both sides of the rope.
T_left - T_right = 0 (Equation 1)
T_left + T_right = 575 N (Equation 2)
To solve this system of equations, we can use the method of substitution or elimination.
Let's choose the method of elimination:
Adding Equation 1 and Equation 2, we get:
2T_left = 575 N
Dividing both sides by 2, we find:
T_left = 287.5 N
Now substitute this value into Equation 2 to find T_right:
287.5 N + T_right = 575 N
Subtracting 287.5 N from both sides, we get:
T_right = 287.5 N
Therefore, the tension in the rope to the left of the mountain climber is 287.5 N, and the tension in the rope to the right of the mountain climber is also 287.5 N.